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• Mountain man - alternate explanation
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  A graphical way of thinking about this is to draw two graphs, both with time (6 am to 6 pm) on the x-axis and place on the mountain on the y-axis.
  
  A steadily upward sloping line on the first graph represents the ascent. A steadily downward sloping line on the second graph represents the descent. Horizontal parts of either line are rests.
  
  Now, superimposing the two graphs shows that the two lines must intersect. The intersection demonstrates that the two paths cross in the same place (y-coordinate) at the same time (x-coordinate).
  
  I don't think the ghost analogy explains why the intersection must occur at the same time of day.

  Posted by walaszek on August 12, 2006 at 13:58:52 Pacific Time

• River Crossing Alternate Solution
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  The first two steps of my solution, contrary to the official answer, were:
  
  

  1. C and C cross
  
  W = { A, A, A, C }
  
  E = { C, C }
  
  
  
  2. C returns
  
  W = { A, A, A, C, C }
  
  E = { C }

  
  
  Essentially the same (it doesn't matter who brings the first cannibal across the river), but I figured if you're an anthropologist, and "the cannibals will follow all directions given to them by the anthropologists," why not minimize your workload? Call it the "lazy antropologist corollary."
  
  Walaszek, I like the concreteness of the graph illustration. I did think, however, that the ghost was an excellent way of illustrating the answer -- in fact, I was surprised that the illustration accompanying the puzzle in the magazine gave away the solution so readily. When the hiker passes his ghost, they are by definition at the same point at the same time. And since they travel opposite routes with identical stop and start times, they must pass each other.

  Posted by TPIRman on August 14, 2006 at 21:42:51 Pacific Time

• Harder version of Jellybeans
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  These were all pretty easy - but there is a neat variation on the Jellybeans one that seems impossible:
  
  This time, the three jars each contain an infinite number of jellybeans and have blank labels. As before, one contains all red, another all blue and the third contains a mixture. What is the least number of beans you have to take out in order that you can correctly write 'Red', 'Blue' or 'Mixed' on the three labels?
  
  WARNING - STOP READING NOW BECAUSE THE ANSWER FOLLOWS:
  
  Four - or three at a stretch...
  
  OK - you read ahead didn't you?
  
  WARNING - STOP READING NOW BECAUSE THE ACTUAL ANSWER FOLLOWS:
  
  Yes - it really is four...or maybe three...First take one bean from each jar. You now have two beans of one colour (call that colour 'A') and one bean of the other colour (call it 'B'). One of the two jars that you got an 'A' from is all 'A' and the other is a mixture of 'A' and 'B' - but we don't know which is which yet. The jar you got the 'B' from must be all 'B' so you can write that on its label right now. But we can never guarantee which of the other two jars is all 'A' and which is the mixture without removing beans from each of the two jars until we find a 'B'. Since there is an infinite number of beans in each jar, we can never empty them and therefore we may never know whether there is a B-coloured bean lurking down at the bottom underneath an infinite sea of A's. So is the problem insoluable? Not at all! Take a second 'B' from the jar of known-B's. You now have two A's and two B's in your hand - pop one of each into each of the two unknown jars - and now you can label both of them as "Mixed" with a clear conscience! You could do it with only three beans by cutting the 'B'-coloured bean in half and putting half into each of the two non-B jars - but cutting up jelly beans without eating them is hard.
  

  Posted by 3D_geek on August 19, 2006 at 03:43:04 Pacific Time